A373242 -id:A373242 - cp's OEIS frontend

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Olivier Gérard, May 29 2024

This difference is always nonnegative.
The number of zero values in each row is A098859.
The number of ones in each row is A325244.
The number of positive entries in each row is A336866.
The corresponding regular triangle for partitions of n of length k is A373242.
The sum of each row is A373243.

			The array begins
  0
  0,0
  0,1,0
  0,1,0,0,0
  0,1,1,0,0,0,0
  0,1,1,0,0,2,0,0,1,0,0
  0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
  0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
  0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
  ...

Olivier Gérard, Table of n, a(n) for n = 1..215307

Cf. A373269 a triangle of the same shape and order for number of multiplicities.

Cf. A080577, A098859, A325244, A336866, A373242, A373243.

Flatten @ Table[
  Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
   IntegerPartitions[n]], {n, 1, 20}]

A373243 a(n) = sum for all integer partitions of n of the difference between number of different parts and number of different multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 6, 11, 18, 27, 36, 61, 77, 115, 161, 223, 291, 416, 531, 729, 951, 1256, 1605, 2132, 2694, 3491, 4423, 5659, 7079, 9027, 11201, 14102, 17484, 21789, 26822, 33309, 40734, 50160, 61195, 74893, 90846, 110722, 133697, 162026, 195104, 235244

Offset: 1

Olivier Gérard, May 29 2024

nonn

Sum of the rows of A373241 or A373242.

			From the eighth row of A373241: a(8)=11
  0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0
or the tenth row of A373242: a(10)=27
  0, 4, 8, 8, 5, 1, 1, 0, 0, 0

Cf. A373241, A373242.

Table[Plus @@
  Table[Plus @@
    Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]

A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 4, 3, 2, 1, 1, 3, 7, 6, 4, 2, 1, 1, 4, 8, 8, 6, 4, 2, 1, 1, 4, 10, 12, 10, 5, 4, 2, 1, 1, 5, 12, 15, 13, 11, 6, 4, 2, 1, 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1, 1, 6, 16, 25, 26, 21, 16, 10, 6, 4, 2, 1, 1, 6, 20, 33, 36, 34, 24, 17, 11, 6, 4, 2, 1, 1, 7, 22, 38, 46, 44, 34, 25, 17, 11, 6, 4, 2, 1, 1, 7, 25, 48, 58, 56, 50, 38, 24, 16, 11, 6, 4, 2, 1

Offset: 1

Olivier Gérard, May 29 2024

			Array begins:
  1,
  1, 1,
  1, 1,  1,
  1, 2,  2,  1,
  1, 2,  4,  2,  1,
  1, 3,  4,  3,  2,  1,
  1, 3,  7,  6,  4,  2,  1,
  1, 4,  8,  8,  6,  4,  2, 1,
  1, 4, 10, 12, 10,  5,  4, 2, 1,
  1, 5, 12, 15, 13, 11,  6, 4, 2, 1,
  1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,
  ...
Example of computation:
T(9,3) = 10 because the partitions of 9 into 3 parts are
  7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
the number of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
and the sum of these multiplicities is 10.

Alois P. Heinz, Rows n = 0..200, flattened (first 40 rows from Olivier Gérard)

Cf. A373242, A373269, A373271.

Flatten@Table[
  Plus @@@
   Table[Map[Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]

A373244 T(n,k) = number of integer partitions of n into k parts for which the number of distinct parts is equal to the number of distinct multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 0, 4, 2, 3, 1, 2, 1, 1, 1, 1, 4, 3, 2, 4, 2, 2, 1, 1, 1, 0, 5, 3, 4, 5, 4, 2, 2, 1, 1, 1, 1, 5, 3, 3, 5, 4, 3, 2, 2, 1, 1, 1, 0, 6, 4, 5, 8, 6, 5, 4, 2, 2, 1, 1, 1, 1, 6, 4, 5, 10, 6, 7, 5, 4, 2, 2, 1, 1

Offset: 1

Olivier Gérard, May 29 2024

Row sum is A098859 (Wilf partitions of n).

Counts the zeros in A373241 or A373242.

			Array begins:
  1,
  1, 1,
  1, 0, 1,
  1, 1, 1, 1,
  1, 0, 2, 1, 1,
  1, 1, 2, 1, 1, 1,
  1, 0, 3, 2, 2, 1, 1,
  1, 1, 3, 2, 2, 2, 1, 1,
  1, 0, 4, 2, 3, 1, 2, 1, 1
  ...

See references listed in A098859.

Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Olivier Gérard)

Flatten[Table[
  Plus @@@
   Table[Count[
     Map[Length[Union[#]] == Length[Union[Length /@ Split[#]]] &,
      IntegerPartitions[n, {k}]], True], {k, 1, n}], {n, 1, 20}]]

cp's OEIS Frontend

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

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A373243 a(n) = sum for all integer partitions of n of the difference between number of different parts and number of different multiplicities.

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Examples

Crossrefs

Programs

Mathematica

A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A373244 T(n,k) = number of integer partitions of n into k parts for which the number of distinct parts is equal to the number of distinct multiplicities.

Views

Author

Keywords

Comments

Examples

References

Links

Programs

Mathematica